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Dirac matrices
The Dirac matrices are a set of 16 matrices created from the Pauli matrices by using the . All 16 Dirac matrices square to positive one (i.e. I_4 ). Any 5 that anticommute can be used as the basis for Cℓ5,0(R'''). Dirac originally used \alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5 which are shown in blue. \begin{array}{c|c|c|c} \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix} & \begin{pmatrix} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&1\\ 0&0&1&0 \end{pmatrix} & \begin{pmatrix} 0&-i&0&0\\ i&0&0&0\\ 0&0&0&-i\\ 0&0&i&0 \end{pmatrix} & \begin{pmatrix} 1&0&0&0\\ 0&-1&0&0\\ 0&0&1&0\\ 0&0&0&-1 \end{pmatrix} \\\hline \begin{pmatrix} 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0 \end{pmatrix} & {\color{blue} \begin{pmatrix} 0&0&0&1\\ 0&0&1&0\\ 0&1&0&0\\ 1&0&0&0 \end{pmatrix} } & {\color{blue} \begin{pmatrix} 0&0&0&-i\\ 0&0&i&0\\ 0&-i&0&0\\ i&0&0&0 \end{pmatrix} } & {\color{blue} \begin{pmatrix} 0&0&1&0\\ 0&0&0&-1\\ 1&0&0&0\\ 0&-1&0&0 \end{pmatrix} } \\\hline {\color{blue} \begin{pmatrix} 0&0&-i&0\\ 0&0&0&-i\\ i&0&0&0\\ 0&i&0&0 \end{pmatrix}} & \begin{pmatrix} 0&0&0&-i\\ 0&0&-i&0\\ 0&i&0&0\\ i&0&0&0 \end{pmatrix} & \begin{pmatrix} 0&0&0&-1\\ 0&0&1&0\\ 0&1&0&0\\ -1&0&0&0 \end{pmatrix} & \begin{pmatrix} 0&0&-i&0\\ 0&0&0&i\\ i&0&0&0\\ 0&-i&0&0 \end{pmatrix} \\\hline {\color{blue} \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{pmatrix}} & \begin{pmatrix} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&-1\\ 0&0&-1&0 \end{pmatrix} & \begin{pmatrix} 0&-i&0&0\\ i&0&0&0\\ 0&0&0&i\\ 0&0&-i&0 \end{pmatrix} & \begin{pmatrix} 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&1 \end{pmatrix} \end{array} Surprisingly the 4 by 4 table above forms a multiplication table even though it is actually created by the following rules: : \begin{array}{c|c|c|c} \sigma_{00} & \sigma_{01} & \sigma_{02} & \sigma_{03} \\\hline \sigma_{10} & {\color{blue} \sigma_{11}} & {\color{blue} \sigma_{12}} & {\color{blue} \sigma_{13}} \\\hline {\color{blue} \sigma_{20}} & \sigma_{21} & \sigma_{22} & \sigma_{23} \\\hline {\color{blue} \sigma_{30}} & \sigma_{31} & \sigma_{32} & \sigma_{33} \end{array} \quad \text{where} \quad \quad\quad\begin{align} \sigma_{ij} &= \sigma_i \otimes \sigma_j \end{align} where \sigma_i and \sigma_j are the original 2x2 Pauli matrices and \otimes is the (not the tensor product) : \sigma_0 = \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}, \quad \sigma_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \quad \sigma_2 = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \quad \sigma_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} The Dirac matrices are commonly referred to by the following name. Note that 4 of the Dirac matrices are denoted \sigma_i even though the same symbol can refer to the original Pauli matrices. : \begin{array}{c|c|c|c} \sigma_0 & \sigma_1 & \sigma_2 & \sigma_3 \\\hline \rho_1 & {\color{blue} \alpha_1} & {\color{blue} \alpha_2} & {\color{blue} \alpha_3} \\\hline {\color{blue} \rho_2} & y_1 & y_2 & y_3 \\\hline {\color{blue} \rho_3} & \delta_1 & \delta_2 & \delta_3 \end{array} \quad \text{where} \quad \begin{align} \sigma_0 &= I_4 \\ -\rho_1 &= y_5 \\ \rho_2 &= {\color{blue} \alpha_5} \\ \rho_3 &= {\color{blue} \alpha_4} = y_4 \end{align} The 16 original Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214). # {\color{blue} \alpha_1, \alpha_2, \alpha_3,} \quad \rho_3, \rho_2 \quad ({\color{blue} \alpha_4, \alpha_5}) # y_1, y_2, y_3, \quad \rho_3, -\rho_1 \quad (y_4, y_5) # \delta_1, \delta_2, \delta_3, \quad \rho_1, \rho_2 # \alpha_1, y_1, \delta_1, \quad \sigma_2, \sigma_3 # \alpha_2, y_2, \delta_2, \quad \sigma_1, \sigma_3 # \alpha_3, y_3, \delta_3, \quad \sigma_1, \sigma_2 Any of the 15 original Dirac matrices (excluding the identity matrix \sigma_0 ) anticommute with eight other original Dirac matrices and commute with the remaining eight, including itself and the identity matrix. :Source: Weisstein, Eric W. "Dirac Matrices." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiracMatrices.html Alpha multiplication table Note that unlike the 3 dimensional case with Pauli matrices, the pseudoscalar in 5 dimensions is the negative of the identity matrix. ( \alpha_1 \alpha_2 \alpha_3 \alpha_4 \alpha_5 = -I_4 = i_2 \otimes i_2. ) In seven dimensions it would be i \cdot I again. See . {\small \begin{array}{r|rrrr} & {\color{blue}1} & {\color{blue}2} & {\color{blue}3} & {\color{blue}4} & {\color{blue}5} \\\hline {\color{blue}1} & & 12 & 13 & 14 & 15 \\ {\color{blue}2} & 21 & & 23 & 24 & 25 \\ {\color{blue}3} & 31 & 32 & & 34 & 35 \\ {\color{blue}4} & 41 & 42 & 43 & & 45 \\ {\color{blue}5} & 51 & 52 & 53 & 54 & \\\hline 21 & {\color{blue}2} & {\color{blue}-1} & 54 & 53 & 43 \\ 31 & {\color{blue}3} & 54 & {\color{blue}-1} &-52 & 42 \\ 32 &-54 & {\color{blue}3} & {\color{blue}-2} & 51 &-41 \\ {\color{green}41} & {\color{blue}4} & 53 & 52 &{\color{blue}-1} &-32 \\ {\color{green}42} & 53 & {\color{blue}4} &-51 &{\color{blue}-2} & 31 \\ {\color{green}43} &-52 & 51 & {\color{blue}4} &{\color{blue}-3} &-21 \\ 51 & {\color{blue}5} & 43 &-42 & 32 &{\color{blue}-1} \\ 52 &-43 & {\color{blue}5} & 41 &-31 &{\color{blue}-2} \\ 53 & 42 &-41 & {\color{blue}5} & 21 &{\color{blue}-3} \\ 54 &-32 & 31 &-21 & {\color{blue}5} &{\color{blue}-4} \end{array}} Gamma matrices The '''gamma matrices, \{ \gamma^0, \gamma^1, \gamma^2, \gamma^3 \} are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra C''ℓ1,3('R''').Wikipedia:Gamma matrices : \begin{align} {\color{blue} \alpha_4 \alpha_0} = {\color{green} \gamma^0} &= {\color{green} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}}, \quad & {\color{blue} \alpha_4 \alpha_1} = {\color{green} \gamma^1} &= {\color{green} \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix}} \\ {\color{blue} \alpha_4 \alpha_2} = {\color{green} \gamma^2} &= {\color{green} \begin{pmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{pmatrix}}, \quad & {\color{blue} \alpha_4 \alpha_3} = {\color{green} \gamma^3} &= {\color{green} \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}} \end{align} The first squares to one and the rest square to negative one: : ({\color{green} \gamma^0})^2 = I_4 \\ ({\color{green} \gamma^1})^2 = -I_4 \\ ({\color{green} \gamma^2})^2 = -I_4 \\ ({\color{green} \gamma^3})^2 = -I_4 For reference: : {\color{blue} \alpha_4 \alpha_5} = \begin{pmatrix} 0 & 0 & -i & 0 \\ 0 & 0 & 0 & -i \\ -i & 0 & 0 & 0 \\ 0 & -i & 0 & 0 \end{pmatrix} References